# Hervé Abdi:

# The Bonferonni and Šidák Corrections

α[PT ] =

number of significant tests total number of tests

=

2, 403 = .0479 . 50, 000

(1)

This value falls close to the theoretical value of α = .05.

For 7,868 families, no test reaches significance. Equivalently for 2,132 families (10,000−7,868) at least one Type I error is made. From these data, α[PF ] can be estimated as:

number of families with at least 1 Type I error α[PF ] = total number of families

=

2, 132 = .2132 . 10, 000

(2)

# This value falls close to the theoretical value of

α[PF ] = 1 − (1 − α[PT ])^{C }= 1 − (1 − .05)^{5 }= .226 .

2.3

How to correct for multiple tests: Šidàk, Bonfer- onni, Boole, Dunn

Recall that the probability of making as least one Type I error for a family of C tests is

α[PF ] = 1 − (1 − α[PT ])^{C }

.

This equation can be rewritten as

α[PT ] = 1 − (1 − α[PF ])^{1/C }.

This formula—derived assuming independence of the tests—is some- times called the Šidàk equation. It shows that in order to reach a given α[PF ] level, we need to adapt the α[PT ] values used for each test.

Because the Šidàk equation involves a fractional power, it is dif- ficult to compute by hand and therefore several authors derived

5